- #1
issacnewton
- 1,035
- 37
Hi
I have to negate the following statements and then express again
in English. I need to know if I am making any mistakes.
a)Everyone who is majoring in math has a friend who needs
help with his homework.
b)Everyone has a roommate who dislikes everyone.
c)There is someone in the freshman class who doesn't
have a roommate.
d)Everyone likes someone,but no one likes everyone.
My answers are as follows--------
a) Let P(x)= x is majoring in math
Q(x)=x needs help with homework
M(x,y)=x and y are friends
The statement would be
\exists x (P(x))\rightarrow \exists y (M(x,y)\wedge Q(y))
So the negated statement would be
\neg \left[\exists x (P(x))\rightarrow \exists y (M(x,y)\wedge Q(y))\right]
\neg \left[\neg(\exists x (P(x)) \vee \exists y (M(x,y)\wedge Q(y))\right]
\neg \left[\forall x \neg P(x) \vee \exists y (M(x,y)\wedge Q(y))\right]
\neg (\forall x \neg P(x)) \wedge \neg \exists y (M(x,y)\wedge Q(y))
\exists x P(x) \wedge \forall y \neg (M(x,y)\wedge Q(y))
\exists x P(x) \wedge \forall y (\neg M(x,y) \vee \neg Q(y))
To translate this statement back to English would be
Some person is majoring in math and everyone is either not a
friend of this person or doesn't need help in homework.
b) let Q(x,y)=x and y are roommates
M(x,y)=x likes y
The statement would be
\forall x \left[\exists y (Q(x,y) \wedge \forall z(\neg M(y,z)))\right]
so the nagated statement would be
\neg \forall x \left[\exists y (Q(x,y)\wedge \forall z(\neg M(y,z)))\right]
\exists x \forall y \left[\neg Q(x,y) \vee \exists z(M(y,z)) \right]
Translation:
Either there is some person who is not roommate with anybody or there is
someone who is liked by all.
c)let P(x)= x is in freshman class.
M(x)=x has a roommate.
The statement would be
\exists x \left[ P(x)\wedge \neg M(x) \right]
So the negated statement is
\neg \exists x \left[ P(x)\wedge \neg M(x) \right]
\forall x \left[ \neg P(x) \vee M(x) \right]
Translation:
Everyone either is not in freshman class or has a roommate.
d)let M(x,y)= x likes y
The statement would be
\forall x \left[ \exists y (M(x,y))\wedge \exists z \neg M(x,z) \right]
The negated statement would be
\neg \forall x \left[ \exists y (M(x,y))\wedge \exists z \neg M(x,z) \right]
\neg \left[\forall x \exists y (M(x,y))\wedge \forall x \exists z \neg M(x,z) \right]
(\neg \forall x \exists y M(x,y)) \vee (\neg \forall x \exists z \neg M(x,z) )
\left[\exists x \forall y \neg M(x,y)\right] \vee \left[\exists x \forall z M(x,z)\right]
Translation:
Either there is someone who likes everyone or there is someone who doesn't like
everyone.
Please comment
Thanks
I have to negate the following statements and then express again
in English. I need to know if I am making any mistakes.
a)Everyone who is majoring in math has a friend who needs
help with his homework.
b)Everyone has a roommate who dislikes everyone.
c)There is someone in the freshman class who doesn't
have a roommate.
d)Everyone likes someone,but no one likes everyone.
My answers are as follows--------
a) Let P(x)= x is majoring in math
Q(x)=x needs help with homework
M(x,y)=x and y are friends
The statement would be
\exists x (P(x))\rightarrow \exists y (M(x,y)\wedge Q(y))
So the negated statement would be
\neg \left[\exists x (P(x))\rightarrow \exists y (M(x,y)\wedge Q(y))\right]
\neg \left[\neg(\exists x (P(x)) \vee \exists y (M(x,y)\wedge Q(y))\right]
\neg \left[\forall x \neg P(x) \vee \exists y (M(x,y)\wedge Q(y))\right]
\neg (\forall x \neg P(x)) \wedge \neg \exists y (M(x,y)\wedge Q(y))
\exists x P(x) \wedge \forall y \neg (M(x,y)\wedge Q(y))
\exists x P(x) \wedge \forall y (\neg M(x,y) \vee \neg Q(y))
To translate this statement back to English would be
Some person is majoring in math and everyone is either not a
friend of this person or doesn't need help in homework.
b) let Q(x,y)=x and y are roommates
M(x,y)=x likes y
The statement would be
\forall x \left[\exists y (Q(x,y) \wedge \forall z(\neg M(y,z)))\right]
so the nagated statement would be
\neg \forall x \left[\exists y (Q(x,y)\wedge \forall z(\neg M(y,z)))\right]
\exists x \forall y \left[\neg Q(x,y) \vee \exists z(M(y,z)) \right]
Translation:
Either there is some person who is not roommate with anybody or there is
someone who is liked by all.
c)let P(x)= x is in freshman class.
M(x)=x has a roommate.
The statement would be
\exists x \left[ P(x)\wedge \neg M(x) \right]
So the negated statement is
\neg \exists x \left[ P(x)\wedge \neg M(x) \right]
\forall x \left[ \neg P(x) \vee M(x) \right]
Translation:
Everyone either is not in freshman class or has a roommate.
d)let M(x,y)= x likes y
The statement would be
\forall x \left[ \exists y (M(x,y))\wedge \exists z \neg M(x,z) \right]
The negated statement would be
\neg \forall x \left[ \exists y (M(x,y))\wedge \exists z \neg M(x,z) \right]
\neg \left[\forall x \exists y (M(x,y))\wedge \forall x \exists z \neg M(x,z) \right]
(\neg \forall x \exists y M(x,y)) \vee (\neg \forall x \exists z \neg M(x,z) )
\left[\exists x \forall y \neg M(x,y)\right] \vee \left[\exists x \forall z M(x,z)\right]
Translation:
Either there is someone who likes everyone or there is someone who doesn't like
everyone.
Please comment
Thanks